Stokesian Dynamics Research Articles

Interactions between hard spheres sedimenting at low Reynolds number

The present article reports on experimental and numerical study for sedimentation of non-colloidal suspended particles in a viscous fluid. The hydrodynamic interactions between several particles sedimenting at low Reynolds numbers have been investigated. The numerical model is based on separation of the real velocity field into two parts. The first part is symmetrical and represents the conventional Stokes contribution. The second non-symmetrical part represents the net inertial contribution. The first contribution has been modelled using the Stokesian Dynamics method. Whereas the second one has been accounted for by assuming the validity of Faxen's first law. The numerical results agree with those from literature in the limit of the Stokes' flow regime ( Re = 0 ) . Unfortunately, when Re > 0 , the simulation appears to quantitatively overestimate the influence of the real inertial effects. However, the accuracy of the results has been observed to be improved by introducing the empirical correction proposed by Happel et al. at the level of the individual particle. In this way, the proposed numerical tool becomes able to determine accurately the behaviour of several particles in different configurations. The agreement between the simulation results and the experimental ones is very satisfactory.

Interpretation of conservative forces from Stokesian dynamic simulations of interfacial and confined colloids

This paper presents Stokesian dynamics simulations of experiments involving one or two charged colloids near either a single charged wall or confined between parallel charged walls. Equilibrium particle-particle and particle-wall interactions are interpreted from dynamic particle trajectories in simulations involving (1) a single particle levitated above a wall, (2) two particles below a wall, and (3) two particles confined between two parallel walls. By specifying only repulsive electrostatic Derjaguin-Landau-Verwey-Overbeek (DLVO) potentials and including multibody hydrodynamics, we successfully recover expected potentials in some cases, while anomalous attraction is observed in other cases. Attraction inferred in the latter simulations displays quantitative agreement with literature measurements when particle dynamics are interpreted using reported analyses. Because anomalous attraction is reproduced in simulations using only electrostatic repulsive DLVO potentials, our results reveal the one-dimensional analyses to be invalid for configurations that are inherently multidimensional via multibody hydrodynamics. Parameters related to experimental sampling of particle dynamics are also found to be critical for obtaining accurate potentials. We explain the anomalous attraction in each experiment using effective potentials, which can be employed in an a priori fashion to assist the confident design of future experiments involving interfacial and confined colloids. Ultimately, our findings reveal the importance of dimensionality and multibody hydrodynamics for understanding nonequilibrium dynamics of colloids near surfaces.

“Contact networks” in continuously shear thickening colloids

Many body effects that occur in continuous shear thickening of concentrated colloid suspensions are examined by using a Stokesian dynamics simulation of model systems of polymer coated particles. The shear thickening state is probed in a number of ways: computed scattering intensities, imaging of density variation using Voronoi constructions, examination of the distribution of interparticle forces, and computation of the fabric of contacts. The shear thickening transition in these systems is found to be associated with the growth of a network of close particle contacts and shear induced density variations. This paper focuses on the network of contacts. The distributions of force magnitude are found to be exponential. The network directly relates to the normal stress differences. Both the data and simple physical argument suggest that thickening can be viewed as an approach to “jamming.”

Continuous shear thickening transitions in model concentrated colloids—The role of interparticle forces

The role of interactions between close particles in the shear thickening of concentrated colloids is examined by using a Stokesian dynamics simulation of model systems. The interactions are repulsive thermodynamic forces and lubrication forces. Three different models are contrasted in their thickening behavior: Brownian spheres, polymer coated spheres, and Hookian spheres. Respectively, they show: a “mild” continuous thickening, a “strong” continuous thickening, and a strain thickening with loss of steady state. The relationship of order-disorder transitions and thickening is examined. Depending on the volume fraction and range of repulsive forces, thickening can be observed with or without an order-disorder transition at its onset. The different thickening responses arise from the dependence of the relaxation time of close particle contacts on interparticle gap. A time-scale based criterion for strong thickening is proposed and supported by the simulations. A simple theoretical model based the motion of a pair of particles leads to this criterion, but also predicts the mild thickening of Brownian spheres. It gives a simple fitting of flow curves which includes the details of the interparticle interactions.

Microstructure and velocity fluctuations in sheared suspensions

The velocity fluctuations present in macroscopically homogeneous suspensions of neutrally buoyant, non-Brownian spheres undergoing simple shear flow, and their dependence on the microstructure developed by the suspensions, are investigated in the limit of vanishingly small Reynolds numbers using Stokesian dynamics simulations. We show that, in the dilute limit, the standard deviation of the velocity fluctuations is proportional to the volume fraction, in both the transverse and the flow directions, and that a theoretical prediction, which considers only for the hydrodynamic interactions between isolated pairs of spheres, is in good agreement with the numerical results at low concentrations. We also simulate the velocity fluctuations that would result from a random hard-sphere distribution of spheres in simple shear flow, and thereby investigate the effects of the microstructure on the velocity fluctuations. Analogous results are discussed for the fluctuations in the angular velocity of the suspended spheres. In addition, we present the probability density functions for all the linear and angular velocity components, and for three different concentrations, showing a transition from a Gaussian to an Exponential and finally to a Stretched Exponential functional form as the volume fraction is decreased. We also show that, although the pair distribution function recovers its fore-aft symmetry in dilute suspensions, it remains anisotropic and that this anisotropy can be accurately described by assuming the complete absence of any permanent doublets of spheres. We finally present a simple correction to the analysis of laser-Doppler velocimetry measurements.

Adaptive Shape Reconfiguration of a Decentralized Motile System Exploiting Molecular Dynamics and Stokesian Dynamics Methods

This paper discusses a fully decentralized algorithm able to create a coherent swarm of autonomous mobile robots from the viewpoint of computational physics. To this end, we focus on Molecular Dynamics and Stokesian Dynamics, both of which are widely used to investigate many-body systems. To verify the feasibility of our approach, this idea has been implemented to a swarm of 2-D radio-connected autonomous mobile robots as a practical example. Simulation results indicate that the proposed algorithm can control the shape of the swarm appropriately based on the current situation without losing the coherence of the swarm nor exchanging global information among modules. Furthermore, we found that local interaction used to exploit Stokesian Dynamics plays an essential role to maintain the coherence of the swarm particularly in an unstructured environment.

Spherical particle in Poiseuille flow between planar walls.

We study a spherical mesoparticle suspended in Newtonian fluid between plane-parallel walls with incident Poiseuille flow. Using a two-dimensional Fourier transform technique we obtain a symmetric analytic expression for the Green tensor for the Stokes equations describing the creeping flow in this geometry. From the matrix elements of the Green tensor with respect to a complete vector harmonic basis, we obtain the friction matrix for the sphere. The calculation of matrix elements of the Green tensor is done in large part analytically, reducing the evaluation of these elements to a one-dimensional numerical integration. The grand resistance and mobility matrices in Cartesian form are given in terms of 13 scalar friction and mobility functions which are expressed in terms of certain matrix elements calculated in the spherical basis. Numerical calculation of these functions is shown to converge well and to agree with earlier numerical calculations based on boundary collocation. For a channel width broad with respect to the particle radius, we show that an approximation defined by a superposition of single-wall functions is reasonably accurate, but that it has large errors for a narrow channel. In the two-wall geometry the friction and mobility functions describing translation-rotation coupling change sign as a function of position between the two walls. By Stokesian dynamics calculations for a polar particle subject to a torque arising from an external field, we show that the translation-rotation coupling induces sideways migration at right angles to the direction of fluid flow.

Shear-induced self-diffusion in non-colloidal suspensions

Self-diffusion in a monodisperse suspension of non-Brownian particles in simple shear flow is studied using accelerated Stokesian dynamics (ASD) simulation. The availability of a much faster computational algorithm allows the study of large systems (typically of 1000 particles) and the extraction of accurate results for the complete shear-induced self-diffusivity tensor. The finite, and often large, autocorrelation time requires the mean-square displacements to be followed for very long times, which is now possible with ASD. The self-diffusivities compare favourably with the available experimental measurements when allowance is made for the finite strains sampled in the experiments. The relationship between the mean-square displacements and the diffusivities appearing in a Fokker–Planck equation when advection couples to diffusion is discussed.

Methods for Stokesian dynamics simulations of nonspherical particles and chains.

The microstructure and properties of suspensions of nonspherical particles are influenced by the specific particle shapes through hydrodynamic interactions. We describe algorithms for Stokesian dynamics simulations of arbitrary shape particles, rigid or flexible, constructed with appropriate constraints among rigid spherical particles whose hydrodynamic mobility is computable by various available schemes, including the one that we recently described [J. Chem. Phys. 112, 2548 (2000)]. The optimal algorithm also provides for rigid attachment among particles during simulation, by aggregation for example. Its implementation for a system with a general combination of internal coordinate constraints (available in a routine from the author) is tested in simulations of sedimentation of spheroids and chains in bounded and unbounded geometries.

Stochastic Mode Reduction for Particle-Based Simulation Methods for Complex Microfluid Systems

We illustrate the stochastic mode reduction procedure as formulated recently by Majda, Timofeyev, and Vanden-Eijnden [Comm. Pure Appl. Math., 54 (2001), pp. 891--974] (MTV) on the equations of motion underlying various particle-based simulation approaches (such as Stokesian dynamics and Brownian dynamics) and the conceptually distinct dissipative particle dynamics (DPD) simulation approaches for complex microfluid systems. The resulting coarse-grained dynamics are compared and contrasted with each other. We show that the stochastic mode reduction procedure provides a way to recover the Smoluchowski dynamics for a standard model of multiple interacting particles in a fluid. The DPD, however, has some subtle aspects which obstruct the application of the stochastic mode reduction procedure. We discuss the mathematical and physical properties of the DPD method that underlie thisdifficulty.

Aggregate formation and collision efficiency in differential settling

A new method of application of Stokesian dynamics, which can efficiently simulate movements of up to 500 particles with interparticle interactions in reasonable computational times, has been developed for the purpose of investigating particle–cluster aggregation in aqueous systems. The method is applied to monodisperse non-Brownian spherical particles aggregating in differential settling, while repulsive colloidal interaction is presumed to be negligible, so that a minimum separation distance can represent the attractive van der Waals force. The final aggregates formed by this algorithm, composed of 300 primary particles, have a common fractal dimension of ∼2.0. The computed collision efficiency, defined as the product of a global and a capture efficiency, is about 5.77×10 −3. This value is significantly larger than the collision efficiency of primary particles colliding with an impermeable solid sphere of the same size as the aggregate, illustrating the important interplay between the permeability and the formation of aggregates.

Stokesian dynamics of nonspherical particles, chains, and aggregates

The microstructure and properties of suspensions of nonspherical particles are influenced by the specific particle shapes through hydrodynamic interactions, but here traditional numerical approaches of solving the Stokes equations are limited to small systems by computational cost, and often to special particle arrangements by symmetry requirements. On the other hand, the analytical development of a hydrodynamic mobility algorithm for Stokesian dynamics (SD) simulations of rigid nonspherical particles is mathematically involved, must be derived for each distinct particle shape needed, and cannot handle deformable particles. Hence we present algorithms for SD simulations of arbitrary shape particles, rigid or flexible, constructed with appropriate constraints among rigid spherical particles whose hydrodynamic mobility is computable by various available schemes, including ours [J. Chem. Phys. 112, 2548 (2000)]. The optimal algorithm also provides for rigid attachment among particles during simulation, by aggregation for example. Its implementation for a system with internal coordinate constraints is tested in simulations of aggregation of spheres and sedimentation of spheroids and chains in bounded and unbounded geometries.

Accelerated Stokesian dynamics: Brownian motion

A new Stokesian dynamics (SD) algorithm for Brownian suspensions is presented. The implementation is based on the recently developed accelerated Stokesian dynamics (ASD) simulation method [Sierou and Brady, J. Fluid Mech. 448, 115 (2001)] for non-Brownian particles. As in ASD, the many-body long-range hydrodynamic interactions are computed using fast Fourier transforms, and the resistance matrix is inverted iteratively, in order to keep the computational cost O(N log N). A fast method for computing the Brownian forces acting on the particles is applied by splitting them into near- and far-field contributions to avoid the O(N3) computation of the square root of the full resistance matrix. For the near-field part, representing the forces as a sum of pairwise contributions reduces the cost to O(N); and for the far-field part, a Chebyshev polynomial approximation for the inverse of the square root of the mobility matrix results in an O(N1.25 log N) computational cost. The overall scaling of the method is thus roughly of O(N1.25 log N) and makes possible the simulation of large systems, which are necessary for studying long-time dynamical properties and/or polydispersity effects in colloidal dispersions. In this work the method is applied to study the rheology of concentrated colloidal suspensions, and results are compared with conventional SD. Also, a faster approximate method is presented and its accuracy discussed.

Development of Effective Stokesian Dynamics Method for Ferromagnetic Colloidal Dispersions (Cluster-based Stokesian Dynamics Method)

We have developed a new Stokesian dynamics (SD) method for nondilute colloidal dispersions, which enables us to reduce drastically the computation time. To verify the validity of the present method, which is called the “cluster-based SD method,” three-dimensional simulations of a ferromagnetic colloidal dispersion have been carried out for a simple shear flow. The correlation function and viscosity have been evaluated to compare the results obtained by the present method with those obtained by the ordinary SD method and by the method of ignoring hydrodynamic interactions between particles. The results obtained here are summarized as follows. The transient properties from an initial state obtained by the present method agree well with those obtained by the ordinary method, even if a radius r clstr, which defines the cluster formation, is taken as a small value such as r clstr=1.2 d ( d is the particle diameter). Also, the equilibrium properties such as the pair correlation function and viscosity obtained by the present cluster-based method are in satisfactory agreement with those obtained by the ordinary SD method. Furthermore, the cluster-based method drastically reduces the computation time to about one-fourteenth to one-seventieth that of the ordinary method. It is clear from these results that the cluster-based SD method is significantly superior to the ordinary SD method for ferromagnetic colloidal dispersions for which a large model system such as N=1000 or 10,000 is indispensable in simulations.

Colloid flow during thickening--a particle level understanding for core-shell particles.

The flow curve of sheared concentrated colloids can show a region of shear thickening. The underlying mechanisms involved in this effect has long been an issue. Recently the author and co-workers have used Stokesian dynamics approximated for high concentrations to simulate models of polymer coated particles in the thickening regime. This work continues the search for a particle level of understanding these systems in the thickening regime. Previous work found that shear thickening is associated with the formation of a network of contacts between polymer coats. Previous work by the author has examined the fabric (geometry) and texture (density distribution) of the contact network. However despite this many-body effect, it was also found that a very simple mean-field type argument at the level of a pair of particles in contact relates the particle interaction laws to the thickening part of the flow curve up to an unknown factor. In the argument this factor is determined by aspects of the destruction of particle contacts. In this paper several new results are reported. The first concerns the definition of the network and its coordination. The paper contrasts systems with different thickness of polymer coats and gives tentative evidence that the network formed sits close to what is known as the iso-static coordination. The second set of issues concerns the analysis of the lifetimes of particle contacts--in effect an examination of the unknown factor and assumptions of the pair argument. A surprising result is that some particle pairs have relatively long lifetimes. Finally, data is reported for the stress fluctuations in the thickening regime.

The Permeability of Synthetic Fractal Aggregates with Realistic Three-Dimensional Structure

The permeability of fractal porous aggregates with realistic three-dimensional structure is investigated theoretically using model aggregates composed of identical spherical primary particles. Synthetic aggregates are generated by several techniques, including a lattice-based method, simulation of aggregation by differential settling and turbulent shear, and the specification of simple cubic structures, resulting in aggregates characterized by the number of primary particles, solid fraction, characteristic radius, and fractal dimension. Stokesian dynamics is used to determine the total hydrodynamic force on and the distribution of velocity within an aggregate exposed to a uniform flow. The aggregate permeability is calculated by comparing these values with the total force and velocity distribution calculated from the Brinkman equation applied locally and to the entire aggregate using permeability expressions from the literature. The relationship between the aggregate permeability and solid fraction is found to be best predicted by permeability expressions based on cylindrical rather than spherical geometrical elements, the latter tending to underestimate the aggregate permeability significantly. The permeability expressions of Jackson and James or Davies provide good estimates of the force on and flow through porous aggregates of known structure. These relationships are used to identify a number of general characteristics of fractal aggregates.

Rheology and microstructure in concentrated noncolloidal suspensions

The rheological behavior of a monodisperse suspension of non-Brownian particles undergoing simple shear flow in the presence of a weak interparticle force is studied using accelerated Stokesian dynamics. The availability of a faster numerical algorithm permits the investigation of larger systems (typically of 512 particles), and accurate results for the suspension viscosity, first and second normal stress differences, and the particle pressure are determined as a function of the volume fraction. The system microstructure, expressed through the pair-distribution function, is also studied and it is demonstrated how the resulting anisotropy in the pair-distribution function is correlated with the suspension non-Newtonian behavior. The ratio of the normal to excess shear stress is found to be an increasing function of the volume fraction, suggesting different volume fraction scalings for different elements of the stress tensor. The relative strength and range of the interparticle force is varied and its effect on the shear and normal stresses is analyzed. Volume fractions above the equilibrium freezing volume fraction (φ≈0.494) are also studied and it is found that the system exhibits a strong tendency to order under flow for volume fractions below the hard-sphere glass transition; limited results for φ=0.60, however, show that the system is again disordered under shear.

Deterministic and stochastic behaviour of non-Brownian spheres in sheared suspensions

The dynamics of macroscopically homogeneous sheared suspensions of neutrally buoyant, non-Brownian spheres is investigated in the limit of vanishingly small Reynolds numbers using Stokesian dynamics. We show that the complex dynamics of sheared suspensions can be characterized as a chaotic motion in phase space and determine the dependence of the largest Lyapunov exponent on the volume fraction ϕ. We also offer evidence that the chaotic motion is responsible for the loss of memory in the evolution of the system and demonstrate this loss of correlation in phase space. The loss of memory at the microscopic level of individual particles is also shown in terms of the autocorrelation functions for the two transverse velocity components. Moreover, a negative correlation in the transverse particle velocities is seen to exist at the lower concentrations, an effect which we explain on the basis of the dynamics of two isolated spheres undergoing simple shear. In addition, we calculate the probability distribution function of the transverse velocity fluctuations and observe, with increasing ϕ, a transition from exponential to Gaussian distributions.The simulations include a non-hydrodynamic repulsive interaction between the spheres which qualitatively models the effects of surface roughness and other irreversible effects, such as residual Brownian displacements, that become particularly important whenever pairs of spheres are nearly touching. We investigate, for very dilute suspensions, the effects of such a non-hydrodynamic interparticle force on the scaling of the particle tracer diffusion coefficients Dy and Dz, respectively, along and normal to the plane of shear, and show that, when this force is very short-ranged, both are proportional to ϕ2 as ϕ → 0. In contrast, when the range of the non-hydrodynamic interaction is increased, we observe a crossover in the dependence of Dy on ϕ, from ϕ2 to ϕ as ϕ → 0. We also estimate that a similar crossover exists for Dz but at a value of ϕ one order of magnitude lower than that which we were able to reach in our simulations.

Microstructure from simulated Brownian suspension flows at large shear rate

Pair microstructure of concentrated Brownian suspensions in simple-shear flow is studied by sampling of configurations from dynamic simulations by the Stokesian Dynamics technique. Simulated motions are three dimensional with periodic boundary conditions to mimic an infinitely extended suspension. Hydrodynamic interactions through Newtonian fluid and Brownian motion are the only physical influences upon the motion of the monodisperse hard-sphere particles. The dimensionless parameters characterizing the suspension are the particle volume fraction and Péclet number, defined, respectively, as φ=(4π/3)na3 with n the number density and a the sphere radius, and Pe=6πηγ̇a3/kT with η the fluid viscosity, γ̇ the shear rate, and kT the thermal energy. The majority of the results reported are from simulations at Pe=1000; results of simulations at Pe=1, 25, and 100 are also reported for φ=0.3 and φ=0.45. The pair structure is characterized by the pair distribution function, g(r)=P1|1(r)/n, where P1|1(r) is the conditional probability of finding a pair at a separation vector r. The structure under strong shearing exhibits an accumulation of pair probability at contact, and angular distortion (from spherical symmetry at Pe=0), with both effects increasing with Pe. Flow simulations were performed at Pe=1000 for eight volume fractions in the range 0.2⩽φ⩽0.585. For φ=0.2–0.3, the pair structure at contact, g(|r|=2)≡g(2), is found to exhibit a single region of strong correlation, g(2)≫1, at points around the axis of compression, with a particle-deficient wake in the extensional zones. A qualitative change in microstructure is observed between φ=0.3 and φ=0.37. For φ⩾0.37, the maximum g(2) lies at points in the shear plane nearly on the x axis of the bulk simple shear flow Ux=γ̇y, while at smaller φ, the maximum g(2) lies near the compressional axis; long-range string ordering is not observed. For φ=0.3 and φ=0.45, g(2)∼Pe0.7 for 1⩽Pe⩽1000, a slower increase than the g(2)∼Pe predicted theoretically for φ≪1 [Brady and Morris, J. Fluid Mech. 348, 143 (1997)]. Spherical harmonic decomposition of g(r) was performed, and for Pe=1000, expansion convergence is found to be nearly complete when harmonics Ylm to the level l=10 are included.

Simulation of structures and their rheological properties in electrorheological fluids

The viscosity of electrorheological (ER) fluids can be changed drastically by imposing an external electric field. We study the relation between microstructures of ER fluids and their viscosity change by performing Stokesian dynamics simulations of a model ER system both in a quiescent state and in a simple steady shear. From large-scale three-dimensional simulations, it is found that (1) under no shear flow there are two principal phases in structural changes: first aggregation of particles into chains oriented along the field direction, and the subsequent slow coalescence of chains into columns, and (2) under a simple steady shear there are three stages in viscosity changes with increasing the field: Newtonian at a weak field, non-Newtonian at a moderate field, and Bingham plastic with yield stress at a high field.